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The Muon Decay Experiment - asimulation

The historical experiment upon which the present simulation is based wasconducted by Rossi and Hall in 1941. The essence of the experiment was tomeasure the number of muons arriving at their detector as a function of positionin the Earth's atmosphere. Muons are elusive subatomic particles that arecreated when cosmic rays hit air molecules high in the upper atmosphere. Muonsare highly unstable and, when produced in the lab on Earth, decay with a half lifeof 2.2 x 10-6 s (micro-seconds). They provide us with a very precise natural clock!

In the original experiment, the flux of muons was measured at a point near thetop of Mount Washington, New Hampshire at about 2000 m altitude and also atthe base of the mountain. Muons are steadily decaying as they travel down to

earth, and knowing the muon's half life, it is quite easy to predict how manymuons should reach the base of the mountain if we know how many are beingdetected at a point 2000 m higher up. Rossi and Hall found the muon flux at thebase of the mountain was much higher than expected - suggesting that thesemuons were living longer! When the time dilation relationship was applied, theresult could be explained if the muons were traveling at 0.994 c.

In the "Lab Frame"

To begin, let's gain some experience with the idea of half-life and how this willrelate to our study of muons. Discuss and answer the following:

1. In your own words, explain what is meant by the term half-life2. Suppose you have prepared a sample of 1000 muons. How many muons

will remain:a. after 2 half-livesb. after 14.6 micro-seconds?

3. A sample that originally contained 1000 muons now contains only 38. Howold is the sample?

4. If muons are produced 90 km above the earth, and assuming that theytravel 0.999c, how much time is required for the muons to reach groundlevel? Express your answer in:

a. micro-secondsb. half-lives

5. If muons are being produced at a rate of 1000/s at 90 km above sea level,estimate the maximum number of muons that should survive the tripignoring any corrections for time dilation.

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6. Derive the following formula: where is the half-life,No is the original number of muons in the sample and N(t) is the number of

muons after time t. (Hint: )7. The following table provides decay information for a sample of muons.

Plot this data in EXCEL and fit an exponential trendline and from thisdetermine the half-life for the muon.

t (mu s) N(t)

0 1010.00

1 742.56

2 559.38

3 422.774 301.33

5 218.83

6 160.38

7 136.86

8 84.45

9 75.33

10 59.36

11 31.99

The Simulation

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1. Our simulationallows usmeasure themuon flux by

moving aprobe tovariousheights in theatmosphere.Do this byclicking onthe probeanddragging.(Press help

on the appletfor moreinformation.)

2. To make areadingpress thestart button (

foundbottom left)and thenpress the

stop button ().

3. The muonflux will bemeasuredand recordedfor you.

4. Start at analtitude of150 km andmake

measurements in 10 kmsteps (finer ifyou wish) asyou movedown toground level.

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5. When youhavecollectedyour data

press thegraph button

( ) thenright-mouseclick in thegraph paneland choosethe generatetable optionto exportyour data

into EXCEL. image only - click here to load the applet on a separate page

Procedure and Analysis

1. You will need to re-arrange your data to enable you to put it into a tablesimilar to the one given in question 7 above. To do this you will need toestimate the time taken for the muons to travel from 90 kms to the heightsat which you measured the muon flux. Since we expect that the muonsare moving at nearly the speed of light we can estimate this time by simply

adopting "c" as the velocity of the muons. Prepare a table that looks like:

Distance Travelled (from 90 km) Time (micro seconds) Muon Flux

0 --- ---

10 --- ---

2. Plot a curve and trendline for your data and from this determine the half-life of the muons that you measured with the probe.

3. You should get a very different half-life for the moving muons than for thelab frame muons (2.2 micro-seconds). Compare the two half-lives; whichis "longer" and by what factor?

Questions

1. Explain how your data illustrates the equation where

http://turing.kingsu.ca/~map/java/applets/muon/muon.htmlhttp://turing.kingsu.ca/~map/java/applets/muon/muon.html

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2. What value for gamma does your measurement for the half life of themoving muons imply?

3. Use your value for gamma to determine the speed of the muons.

Literature links

http://www.heliwave.com/gaasenbeek/spap5.html

http://www.marxists.org/reference/archive/einstein/works/1910s/relative/index.htm

http://www.heliwave.com/gaasenbeek/spap5.htmlhttp://www.marxists.org/reference/archive/einstein/works/1910s/relative/index.htmhttp://www.heliwave.com/gaasenbeek/spap5.htmlhttp://www.marxists.org/reference/archive/einstein/works/1910s/relative/index.htm